Can solving an exponential function predict the future? Can exponential functions be a fast track to wealth?

As a matter of fact, yes! Successful entrepreneurs all over the world know the value of modeling population and economic growth and decay. By constructing an accurate exponential equation to model the past and present growth of a particular area of the country, state, or even town, the ideal location for a new business to thrive can be predicted.

Some things cannot be modeled of course, such as natural disasters and the like, but often a few well educated guesses make the difference between stunning success and dismal failure in the business world.

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Guidance

In this lesson you will learn about
exponential functions,
a family of functions different from the other function families because the variable x is in the exponent. For example, the functions
f
(
x
) = 2
x
and
g
(
x
) = 100(2)5
x
are exponential functions.

Evaluating Exponential Functions

Consider the function
f
(
x
) = 2
x
. When we input a value for
x
, we find the function value by raising 2 to the exponent of
x
. For example, if
x
= 3, we have
f
(3) = 2
3
= 8.

If we choose larger values of
x
, we will get larger function values, as the function values will be larger powers of 2. For example,
f
(10) = 2
10
= 1,024.

If we choose smaller values of
x
, we will quickly end up with fractions. For instance, if
x = 0
, we have
f
(0) = 2
0
= 1. If
x
= -3, we have
. If we choose smaller and smaller
x
values, the function values will be smaller and smaller fractions. For example, if
x
= -10, we have
. Notice that none of the
x
values we choose will result in a function value of "0", because the numerator of the fraction will always be 1. This tells us that while the domain of this function is the set of all real numbers, the range is limited to the set of positive real numbers.

In general, if we have a function of the form
f
(
x
) = a
x
, where
a
is a positive real number, the domain of the function is the set of all real numbers, and the range is limited to the set of positive real numbers. This restricted domain will result in a specific shape of the graph.

Solving exponential equations

Solving an exponential equation means determining the value of
x
for a given function value. The solution to the equation 2
x
= 8 is the value of
x
that makes the equation a true statement, therefor
x
= 3, since 2
3
= 8.

Example A

Solve for
x
: 3 (2
x
+ 1
) = 24.

Solution

We can solve this equation by writing both sides of the equation as a power of 2:

To solve the equation now, recall a property of exponents: if
b
x
=
b
y
, then
x
=
y
. That is, if two powers of the same base are equal, the exponents must be equal. This property tells us how to solve:

Example B

The values of the function
g
(
x
) = 3
x
behave much like those of
f
(
x
) = 2
x
: if we choose larger values, we get larger and larger function values. If
x
= 0, the function value is 1. And, if we choose smaller and smaller
x
values, the function values will be smaller and smaller fractions. Also, the range of
g
(
x
) is limited to positive values.

Example C

Use a graphing utility to solve each equation:

a. 2
3
x
- 1
= 7

b. 6
-4
x
= 2
8
x
- 5

Solution:

a. 2
3
x
- 1
= 7

Graph the function
y
= 2
3
x
- 1
and find the point where the graph intersects the horizontal line
y
= 7. The solution is
x
≈ 1.27